Cartesian Product (cross product)

Written as A \times B. Every element a \in A gets paired with every element b \in B.

Example: Let A = \{ a_{1}, a_{2}, a_{3} \} and B = \{ b_{1}, b_{2} \}, then: \begin{align} A \times B &= \{ (a_{1}, b_{1}), (a_{1}, b_{2}), (a_{2}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{1}), (a_{3}, b_{2}) \} \\ B \times A & = \{ (b_{1}, a_{1}), (b_{1}, a_{3}), (b_{1}, a_{3}), (b_{2}, a_{1}), (b_{2}, a_{2}), (b_{2}, a_{3})\} \\ B \times B &= \{ (b_{1}, b_{1}), (b_{1}, b_{2}), (b_{2}, b_{1}), (b_{2}, b_{2}) \} \end{align}

Properties

• With A and B being finite: |A\times B| = |A| \cdot |B|
• Not commutative: A \times B \neq B \times A
• Bijection: A\times B \to B \times A, (a,b) \to (b,a)